Rigging A Character With Boolean Modifier And SubSurf?
Part 1: You Can

You can leave the modifiers on without applying them, sure. You would need each of those pieces of the object named "Cube" to be separated into individual objects. Then you would parent each of those objects to your armature bones.

I would disable the Boolean Modifier for the viewport only, because it's killing your viewport performance. 



Then you should be able to navigate and rig smoothly.

Part 2: But in this case you shouldn't

I realize it is tempting to try to carve out pieces of a mesh using Booleans because if you were sculting in the real world with clay or something you would carve away at your shape. But mesh modeling doesn't work that way and Booleans don't create pretty topology. They are very useful and appropriate for certain cases, but this is not such a case.

The best thing to do would be to model your character's topology to have cuts in those places, using quads instead of tris to the best of your ability. The subject of topology is broad, but worth studying if you want to animate characters.

If the places where the Boolean cuts into the mesh are only for a specific scene - like if this character is going to be cut into a bunch of pieces - then after you've modeled your finished character, duplicate a copy of the character and cut that one apart.

Without knowing exactly what your plans are for animating this character, that is the best advice I can give.

On a final note, it looks like you are using an Edge Split Modifier to fix a shading issue, however this is only adding to the heaviness of your scene by creating more geometry. It would be better to fix the ugly shading at Boolean intersections using Auto Smooth, like so: